Gottfried Wilhelm Leibniz was the last schoolman and the first modern logician. He invented different types of formalization of Aristotelian logic which in parts anticipated George Boole's calculus. Most of Leibniz' papers had not been published until about 1900, thus he has exerted only little influence on the development of modern logic. The papers in this section deal with his calculus of characteristic numbers which he invented in April 1679. It is a complete arithmetization of Aristotelian logic in its intensional interpretation. The characteristic numbers have been used by Lukasiewicz in his formalisation of Aristotelian logic and may have been a source of inspiration for Gödel in his famous arithmization of predicate logic.

• [PDF]An Intensional Leibniz Semantics for Aristotelian Logic (The Review of Symbolic Logic, Published online by Cambridge University Press 17 Mar 2010 doi:10.1017/S1755020309990396)
Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standardsemantics of Aristotelian logic considers terms as standing for sets of individuals. From aphilosophical standpoint, this extensional model poses problems: There exist serious doubts thatAristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classicalphilosophy up to Leibniz and Kant had a different view on this question—they looked at termsas standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelianlogic containing a calculus of natural deduction, while, with respect to semantics, he still madeuse of an extensional interpretation. In this paper we deal with a simple intensional semantics forCorcoran’s syntax—intensional in the sense that no individuals are needed for the construction of acomplete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excludingother, “higher” concepts—corresponding to the idea which Leibniz used in the construction of hischaracteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formalsyntax with an adequate semantics which is free from presuppositions which have entered intomodern interpretations of Aristotle’s theory via predicate logic.

• On Leibniz' characteristic numbers
  Studia Leibnitiana Band 43/2002, Seite 161
  147 k [PDF] Opens new window


• G. W. Leibniz - die Utopie der Denkmaschine
  228 k [PDF](in German) Opens new window


• On Negation in Leibniz' System of Charachteristic numbers
  221 k [PDF]. Opens new window
In the spring of 1679, Leibniz invented a famous interpretation ofAristotelian logic by means of his characteristic numbers.Leibniz was able to show that, within his model, all classicallaws of "positive" Aristotelian logic (a term logic withoutnegation) hold, if one uses a certain proper arithmeticalinterpretation of the Aristotelian quantors A, E, I, andO. While this construction of characteristic numbers is a highlyesteemed result today, Leibniz himself was apparently not content withhis achievement. His last notes on this subject show how hard hestruggled with different unsuccessful attempts of allowing alsofor term negation within his formalism. Later on he never resumedhis work on characteristic numbers.
By proving a negative result on characteristic numbers we show in this paper why Leibnizwas bound to fail, and we also demonstrate how to enlarge his system in order to include term negation in a formal correct way.